The First 2 Layers

These are the 41 distinct cases (plus one solved) for the
placement of a corner and its matching edge from the F2L and the
algorithms I use for them. It is highly recommended that you
learn how to do F2L intuitively so that you can take advantage of empty
slots, nonstandard F2L pairings, etc. However, I have provided these
"algorithms" anyway. For most of these cases, I have broken the
algorithms down into two
stages: In the first stage, you will be pairing up the corner and the
edge and AUF so that they can be placed. This stage
will be shown in red.
In
the second stage, the pair will be inserted into the slot in which it
belongs. This stage will be shown in green.
Some
cases do not have pairing-up or insertion stages and will be shown in
the normal black font.

These algorithms appear EXACTLY as I perform them when I am
solving the first two layers, in speedcubing notation with rotations
included in the algorithm. It should be noted that
these are the algorithms that I find easiest to
perform. However, you may find other algorithms
better-suited for your own hands, so it is recommended to try many
different algorithms for the same situation to find which one works
best for your own style of cubing.

I have included several algorithms for each case. The FR, FL,
BR, and BL before an algorithm indicate which F2L slot the pair will be
inserted into. For each of those cases, you would have to rotate the
given image so that the missing F2L pair is in the slot I indicated.

For a printable page of these algorithms, visit my printable page. Please note that you will need Adobe Reader to access and print the printable page.

Corner Correct, Edge Not Placed

In these cases, the corner is already solved and only the edge needs to
be placed. In Cases #02 and #03, it is sometimes
helpful to use a "working corner" to place the edge
piece. This means you could rotate the D-layer so
that an empty corner is under the edge you need to solve, place the
edge with a quick trigger, and finally restore your
D-layer. It is also possible to do this for Case #01
by taking the edge out (with an empty corner), AUF, and then correctly
placing the edge piece.

Separate
the corner from the edge by placing them in the
U-layer. Next, AUF and bring the corner to the
bottom using the empty slot so you can move the edge freely to match-up
with the corner. AUF again and place the block in
the slot it belongs.

Edge Correct, Corner Slot Free

In this group, the edge is already solved, but the corner is in the
U-layer. In Cases #05 and #06, it is sometimes
useful to use a "working edge" to place the
corner. This is very similar to a working corner
method, but you place the corner using a free edge slot instead.

In this
case, the edge is solved, but the corner is not. A
common way of solving this involves removing the entire first layer,
breaking away the corner to match it with its edge, and then restoring
everything.

One way
of doing this is to break up these two pieces and twist the corner so
that the two pieces are opposite each other and then placing
them. The other way of fixing htis case is to just
use the clever algorithm that doesn't necessarily have pair-up and
insertion phases.

31

FR: (R
U R')
BL: (L U L')

Here,
the corner and edge are "opposite" each other and can be placed with a
simple trigger. The goal of many F2L algorithms is
to get the corner and edge into this case and then solve them with this
trigger.